This is a Phase II study to determine the toxicity and tolerance of sequential therapy with cisplatin and prolonged topotecan administration (IV or oral)(C-PORT, followed by paclitaxel plus carboplatin in women with Stage Ic, II, III, or IV epithelial ovarian cancer, previously untreated with chemotherapy. Response rate and time to progression for this therapy will be determined. The study will also determine the relative pharmacokinetics of intravenous infusion and prolonged oral administration of topotecan and compare the pharmacodynamics of topo-1 inhibition when given by the IV or oral route.